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EEE 498/598 Overview of Electrical Engineering. Lecture 2: Introduction to Electromagnetic Fields; Maxwell’s Equations; Electromagnetic Fields in Materials; Phasor Concepts; Electrostatics: Coulomb’s Law, Electric Field, Discrete and Continuous Charge Distributions; Electrostatic Potential.

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eee 498 598 overview of electrical engineering

EEE 498/598Overview of Electrical Engineering

Lecture 2:

Introduction to Electromagnetic Fields;

Maxwell’s Equations; Electromagnetic Fields in Materials; Phasor Concepts;

Electrostatics: Coulomb’s Law, Electric Field, Discrete and Continuous Charge Distributions; Electrostatic Potential

1

lecture 2 objectives
Lecture 2 Objectives
  • To provide an overview of classical electromagnetics, Maxwell’s equations, electromagnetic fields in materials, and phasor concepts.
  • To begin our study of electrostatics with Coulomb’s law; definition of electric field; computation of electric field from discrete and continuous charge distributions; and scalar electric potential.

2

introduction to electromagnetic fields
Introduction to Electromagnetic Fields
  • Electromagnetics is the study of the effect of charges at rest and charges in motion.
  • Some special cases of electromagnetics:
    • Electrostatics: charges at rest
    • Magnetostatics: charges in steady motion (DC)
    • Electromagnetic waves: waves excited by charges in time-varying motion

3

introduction to electromagnetic fields4

Circuit

Theory

Introduction to Electromagnetic Fields

Maxwell’s

equations

Fundamental laws of classical electromagnetics

Geometric Optics

Electro-statics

Magneto-statics

Electro-magnetic waves

Special cases

Statics:

Transmission

Line

Theory

Input from other disciplines

Kirchoff’s Laws

4

introduction to electromagnetic fields5
Introduction to Electromagnetic Fields
  • transmitter and receiver
  • are connected by a “field.”

5

introduction to electromagnetic fields6

2

3

1

4

Introduction to Electromagnetic Fields

High-speed, high-density digital circuits:

  • consider an interconnect between points “1” and “2”

6

introduction to electromagnetic fields7
Introduction to Electromagnetic Fields
  • Propagation delay
  • Electromagnetic coupling
  • Substrate modes

7

introduction to electromagnetic fields8
Introduction to Electromagnetic Fields
  • When an event in one place has an effect on something at a different location, we talk about the events as being connected by a “field”.
  • A fieldis a spatial distribution of a quantity; in general, it can be either scalar or vector in nature.

8

introduction to electromagnetic fields9
Introduction to Electromagnetic Fields
  • Electric and magnetic fields:
    • Are vector fields with three spatial components.
    • Vary as a function of position in 3D space as well as time.
    • Are governed by partial differential equations derived from Maxwell’s equations.

9

introduction to electromagnetic fields10
Introduction to Electromagnetic Fields
  • A scalaris a quantity having only an amplitude (and possibly phase).
  • A vectoris a quantity having direction in addition to amplitude (and possibly phase).

Examples: voltage, current, charge, energy, temperature

Examples: velocity, acceleration, force

10

introduction to electromagnetic fields11
Introduction to Electromagnetic Fields
  • Fundamental vector field quantities in electromagnetics:
    • Electric field intensity
    • Electric flux density (electric displacement)
    • Magnetic field intensity
    • Magnetic flux density

units = volts per meter (V/m = kg m/A/s3)

units = coulombs per square meter (C/m2 = A s /m2)

units = amps per meter (A/m)

units = teslas = webers per square meter (T = Wb/ m2 = kg/A/s3)

11

introduction to electromagnetic fields12
Introduction to Electromagnetic Fields
  • Universal constants in electromagnetics:
    • Velocity of an electromagnetic wave (e.g., light) in free space (perfect vacuum)
    • Permeability of free space
    • Permittivity of free space:
    • Intrinsic impedance of free space:

12

introduction to electromagnetic fields13
Introduction to Electromagnetic Fields
  • Relationships involving the universal constants:

In free space:

13

introduction to electromagnetic fields14

sources

Ji, Ki

fields

E, H

Introduction to Electromagnetic Fields
  • Obtained
  • by assumption
  • from solution to IE

Solution to

Maxwell’s equations

Observable quantities

14

maxwell s equations
Maxwell’s Equations
  • Maxwell’s equations in integral form are the fundamental postulates of classical electromagnetics - all classical electromagnetic phenomena are explained by these equations.
  • Electromagnetic phenomena include electrostatics, magnetostatics, electromagnetostatics and electromagnetic wave propagation.
  • The differential equations and boundary conditions that we use to formulate and solve EM problems are all derived from Maxwell’s equations in integral form.

15

maxwell s equations16
Maxwell’s Equations
  • Various equivalence principles consistent with Maxwell’s equations allow us to replace more complicated electric current and charge distributions with equivalent magnetic sources.
  • These equivalent magnetic sources can be treated by a generalization of Maxwell’s equations.

16

maxwell s equations in integral form generalized to include equivalent magnetic sources
Maxwell’s Equations in Integral Form (Generalized to Include Equivalent Magnetic Sources)

Adding the fictitious magnetic source terms is equivalent to living in a universe where magnetic monopoles (charges) exist.

17

continuity equation in integral form generalized to include equivalent magnetic sources
Continuity Equation in Integral Form (Generalized to Include Equivalent Magnetic Sources)
  • The continuity equations are implicit in Maxwell’s equations.

18

contour surface and volume conventions

S

C

dS

S

V

dS

Contour, Surface and Volume Conventions
  • open surface S bounded by
  • closed contour C
  • dS in direction given by
  • RH rule
  • volume V bounded by
  • closed surface S
  • dS in direction outward
  • from V

19

electric current and charge densities
Electric Current and Charge Densities
  • Jc = (electric) conduction current density (A/m2)
  • Ji = (electric) impressed current density (A/m2)
  • qev = (electric) charge density (C/m3)

20

magnetic current and charge densities
Magnetic Current and Charge Densities
  • Kc = magnetic conduction current density (V/m2)
  • Ki = magnetic impressed current density (V/m2)
  • qmv = magnetic charge density (Wb/m3)

21

maxwell s equations sources and responses
Maxwell’s Equations - Sources and Responses
  • Sources of EM field:
    • Ki, Ji, qev, qmv
  • Responses to EM field:
    • E, H, D, B, Jc, Kc

22

maxwell s equations in differential form generalized to include equivalent magnetic sources
Maxwell’s Equations in Differential Form (Generalized to Include Equivalent Magnetic Sources)

23

continuity equation in differential form generalized to include equivalent magnetic sources
Continuity Equation in Differential Form (Generalized to Include Equivalent Magnetic Sources)
  • The continuity equations are implicit in Maxwell’s equations.

24

surface current and charge densities
Surface Current and Charge Densities
  • Can be either sources of or responses to EM field.
  • Units:
    • Ks - V/m
    • Js - A/m
    • qes - C/m2
    • qms - W/m2

27

electromagnetic fields in materials
Electromagnetic Fields in Materials
  • In time-varying electromagnetics, we consider E and H to be the “primary” responses, and attempt to write the “secondary” responses D, B, Jc, and Kcin terms of E and H.
  • The relationships between the “primary” and “secondary” responses depends on the medium in which the field exists.
  • The relationships between the “primary” and “secondary” responses are called constitutive relationships.

28

electromagnetic fields in materials29
Electromagnetic Fields in Materials
  • Most general constitutive relationships:

29

electromagnetic fields in materials30
Electromagnetic Fields in Materials
  • In free space, we have:

30

electromagnetic fields in materials31
Electromagnetic Fields in Materials
  • In a simple medium, we have:
  • linear(independent of field strength)
  • isotropic(independent of position within the medium)
  • homogeneous(independent of direction)
  • time-invariant(independent of time)
  • non-dispersive (independent of frequency)

31

electromagnetic fields in materials32
Electromagnetic Fields in Materials
  • e = permittivity = ere0 (F/m)
  • m = permeability = mrm0(H/m)
  • s = electric conductivity = ere0(S/m)
  • sm = magnetic conductivity = ere0(W/m)

32

phasor representation of a time harmonic field
Phasor Representation of a Time-Harmonic Field
  • A phasor is a complex number representing the amplitude and phase of a sinusoid of known frequency.

phasor

frequency domain

time domain

33

phasor representation of a time harmonic field34
Phasor Representation of a Time-Harmonic Field
  • Phasors are an extremely important concept in the study of classical electromagnetics, circuit theory, and communications systems.
  • Maxwell’s equations in simple media, circuits comprising linear devices, and many components of communications systems can all be represented as linear time-invariant (LTI) systems. (Formal definition of these later in the course …)
  • The eigenfunctions of any LTI system are the complex exponentials of the form:

34

phasor representation of a time harmonic field35
If the input to an LTI system is a sinusoid of frequency w, then the output is also a sinusoid of frequency w (with different amplitude and phase).Phasor Representation of a Time-Harmonic Field

A complex constant (for fixed w); as a function of w gives the frequency response of the LTI system.

35

phasor representation of a time harmonic field36
Phasor Representation of a Time-Harmonic Field
  • The amplitude and phase of a sinusoidal function can also depend on position, and the sinusoid can also be a vector function:

36

phasor representation of a time harmonic field37
Phasor Representation of a Time-Harmonic Field
  • Given the phasor (frequency-domain) representation of a time-harmonic vector field, the time-domain representation of the vector field is obtained using the recipe:

37

phasor representation of a time harmonic field38
Phasor Representation of a Time-Harmonic Field
  • Phasors can be used provided all of the media in the problem are linear no frequency conversion.
  • When phasors are used, integro-differential operators in time become algebraic operations in frequency, e.g.:

38

time harmonic maxwell s equations
Time-Harmonic Maxwell’s Equations
  • If the sources are time-harmonic (sinusoidal), and all media are linear, then the electromagnetic fields are sinusoids of the same frequency as the sources.
  • In this case, we can simplify matters by using Maxwell’s equations in the frequency-domain.
  • Maxwell’s equations in the frequency-domain are relationships between the phasor representations of the fields.

39

electrostatics as a special case of electromagnetics

Circuit

Theory

Electrostatics as a Special Case of Electromagnetics

Maxwell’s

equations

Fundamental laws of classical electromagnetics

Geometric Optics

Electro-statics

Magneto-statics

Electro-magnetic waves

Special cases

Statics:

Transmission

Line

Theory

Input from other disciplines

Kirchoff’s Laws

42

electrostatics
Electrostatics
  • Electrostaticsis the branch of electromagnetics dealing with the effects of electric charges at rest.
  • The fundamental law of electrostatics is Coulomb’s law.

43

electric charge
Electric Charge
  • Electrical phenomena caused by friction are part of our everyday lives, and can be understood in terms of electrical charge.
  • The effects of electrical charge can be observed in the attraction/repulsion of various objects when “charged.”
  • Charge comes in two varieties called “positive” and “negative.”

44

electric charge45
Electric Charge
  • Objects carrying a net positive charge attract those carrying a net negative charge and repel those carrying a net positive charge.
  • Objects carrying a net negative charge attract those carrying a net positive charge and repel those carrying a net negative charge.
  • On an atomic scale, electrons are negatively charged and nuclei are positively charged.

45

electric charge46
Electric Charge
  • Electric charge is inherently quantized such that the charge on any object is an integer multiple of the smallest unit of charge which is the magnitude of the electron charge e= 1.602  10-19C.
  • On the macroscopic level, we can assume that charge is “continuous.”

46

coulomb s law
Coulomb’s Law
  • Coulomb’s law is the “law of action” between charged bodies.
  • Coulomb’s law gives the electric force between two point chargesin an otherwise empty universe.
  • A point charge is a charge that occupies a region of space which is negligibly small compared to the distance between the point charge and any other object.

47

coulomb s law48
Coulomb’s Law

Q1

Q2

Unit vector in

direction of R12

Force due to Q1

acting on Q2

48

coulomb s law49
Coulomb’s Law
  • The force on Q1due to Q2 is equal in magnitude but opposite in direction to the force on Q2 due to Q1.

49

electric field

Qt

Q

Electric Field
  • Consider a point charge Q placed at the origin of a coordinate system in an otherwise empty universe.
  • A test charge Qtbrought near Qexperiences a force:

50

electric field51
Electric Field
  • The existence of the force on Qt can be attributed to an electric field produced by Q.
  • The electric field produced by Q at a point in space can be defined as the force per unit charge acting on a test charge Qtplaced at that point.

51

electric field52
Electric Field
  • The electric field describes the effect of a stationary charge on other charges and is an abstract “action-at-a-distance” concept, very similar to the concept of a gravity field.
  • The basic units of electric field are newtons per coulomb.
  • In practice, we usually use volts per meter.

52

electric field53
Electric Field
  • For a point charge at the origin, the electric field at any point is given by

53

electric field54

P

Q

O

Electric Field
  • For a point charge located at a point P’ described by a position vector

the electric field at P is given by

54

electric field55
Electric Field
  • In electromagnetics, it is very popular to describe the source in terms of primed coordinates, and the observation point in terms of unprimed coordinates.
  • As we shall see, for continuous source distributions we shall need to integrate over the source coordinates.

55

electric field56
Electric Field
  • Using the principal of superposition, the electric field at a point arising from multiple point charges may be evaluated as

56

continuous distributions of charge
Continuous Distributions of Charge
  • Charge can occur as
    • point charges (C)
    • volume charges (C/m3)
    • surface charges (C/m2)
    • line charges (C/m)

 most general

57

continuous distributions of charge58
Continuous Distributions of Charge
  • Volume charge density

Qencl

DV’

58

continuous distributions of charge59

dV’

P

Qencl

V’

Continuous Distributions of Charge
  • Electric field due to volume charge density

59

continuous distributions of charge61
Continuous Distributions of Charge
  • Surface charge density

Qencl

DS’

61

continuous distributions of charge62
Continuous Distributions of Charge
  • Electric field due to surface charge density

dS’

P

Qencl

S’

62

continuous distributions of charge64
Continuous Distributions of Charge
  • Line charge density

DL’

Qencl

64

continuous distributions of charge65
Continuous Distributions of Charge
  • Electric field due to line charge density

DL’

Qencl

P

65

electrostatic potential
Electrostatic Potential
  • An electric field is a force field.
  • If a body being acted on by a force is moved from one point to another, then workis done.
  • The concept of scalar electric potential provides a measure of the work done in moving charged bodies in an electrostatic field.

67

electrostatic potential68

b

a

q

Electrostatic Potential
  • The work done in moving a test charge from one point to another in a region of electric field:

68

electrostatic potential69

b

a

x

3

5

Electrostatic Potential
  • In evaluating line integrals, it is customary to take the dlin the direction of increasing coordinate value so that the manner in which the path of integration is traversed is unambiguously determined by the limits of integration.

69

electrostatic potential70

C

Electrostatic Potential
  • The electrostatic field is conservative:
    • The value of the line integral depends only on the end points and is independent of the path taken.
    • The value of the line integral around any closed path is zero.

70

electrostatic potential71
Electrostatic Potential
  • The work done per unit charge in moving a test charge from point a to point b is the electrostatic potential difference between the two points:

electrostatic potential difference

Units are volts.

71

electrostatic potential72
Electrostatic Potential
  • Since the electrostatic field is conservative we can write

72

electrostatic potential73
Electrostatic Potential
  • Thus the electrostatic potentialV is a scalar field that is defined at every point in space.
  • In particular the value of the electrostatic potential at any point P is given by

reference point

73

electrostatic potential74
Electrostatic Potential
  • The reference point (P0) is where the potential is zero (analogous to ground in a circuit).
  • Often the reference is taken to be at infinity so that the potential of a point in space is defined as

74

electrostatic potential and electric field
Electrostatic Potential and Electric Field
  • The work done in moving a point charge from point a to pointb can be written as

75

electrostatic potential and electric field76
Electrostatic Potential and Electric Field
  • Along a short path of length Dl we have

76

electrostatic potential and electric field77
Electrostatic Potential and Electric Field
  • Along an incremental path of length dlwe have
  • Recall from the definition of directional derivative:

77

electrostatic potential and electric field78
Electrostatic Potential and Electric Field
  • Thus:

the “del” or “nabla” operator

78